Beltrami's summation formula (see [1]}) :
Let \(0<b<a.\)  Then
\(\forall \theta \in \mathbb {R},\quad K_{0}\left(\sqrt{a^{2}+b^{2}-2ab\cos (\theta )}\right)=\sum _{m=-\infty }^{\infty }I_{m}(b)K_{m}(a)\cos (m\theta ).\) 
